Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #496225 |
| commutator | C22×D4 | C23×C4 |
| Frattini, cosocle | C2×#127 | C24 |
| prime radical | #6437451 | C2 |
| LCS | G ▷ C22×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C24×C4 ◁ C2×#3449 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
3632 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,8)(4,18,20,9)(5,22,26,37,25,36,27,10)(6,12)(7,11)(13,28)(14,39,41,29)(15,43,45,30,23,53,46,57)(16,31)(17,38)(19,49,50,48,24,54,52,33)(21,35,51,34)(32,47)(40,60,61,55,44,64,63,59)(42,56,62,58);(1,3)(2,8)(4,14)(5,23)(6,17)(7,16)(9,29)(10,30)(11,31)(12,38)(13,32)(15,25)(18,39)(19,44)(20,41)(21,42)(22,53)(24,40)(26,46)(27,45)(28,47)(33,55)(34,58)(35,56)(36,43)(37,57)(48,59)(49,64)(50,63)(51,62)(52,61)(54,60);(1,4)(2,9)(3,14)(5,24)(6,21)(7,20)(8,29)(10,33)(11,18)(12,34)(13,35)(15,44)(16,41)(17,42)(19,25)(22,54)(23,40)(26,50)(27,52)(28,51)(30,55)(31,39)(32,56)(36,49)(37,48)(38,58)(43,64)(45,61)(46,63)(47,62)(53,60)(57,59);(1,5,7,25)(2,10,11,37)(3,15,16,23)(4,19,20,24)(6,26,28,27)(8,30,31,57)(9,33,18,48)(12,36,13,22)(14,40,41,44)(17,45,47,46)(21,52,51,50)(29,55,39,59)(32,53,38,43)(34,49,35,54)(42,63,62,61)(56,60,58,64)Polycyclic presentation:
2^10|a^2=g,d^2=j,g^2=ij,h^2=j,c^a=ce,d^a=def,d^b=dei,d^c=dei,e^a=ej,e^d=ej,f^a=fh,f^b=fj,f^d=fj,g^c=gj,g^d=gh,g^f=gij,h^a=hi,h^d=hj,h^f=hj,i^d=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72306877, #72307002, #72317584, … |
| maximal normal subgroup of | #72306877, #72307002, #72317584, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.7);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.5*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.5*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.8);;
gap> SetConjugate(coll, 7, 6, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);